A structured and constrained Total Least-Squares solution with cross-covariances

A new proof is presented of the desirable property of the weighted total least-squares (WTLS) approach in preserving the structure of the coefficient matrix in terms of the functional independent elements. The WTLS considers the full covariance matrix of observed quantities in the observation vector and in the coefficient matrix; possible correlation between entries in the observation vector and the coefficient matrix are also considered. The WTLS approach is then equipped with constraints in order to produce the constrained structured TLS (CSTLS) solution. The proposed approach considers the correlation between the observation vector and the coefficient matrix of an Error-In-Variables model, which is not considered in other, recently proposed approaches. A rigid transformation problem is done by preservation of the structure and satisfying the constraints simultaneously.